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Spectral Asymptotics of Toeplitz Operators on Zoll Manifolds
Journal of Functional Analysis FU3044 journal of functional analysis 146, 496516 (1997) article no. FU963044
Spectral Asymptotics of Toeplitz Operators on Zoll Manifolds V. Guillemin* and K. Okikiolu Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received June 20, 1996; revised August 30, 1996; accepted September 2, 1996
We prove a generalization of the Strong Szego Limit Theorem for Zoll type operators on smooth compact closed manifolds. More precisely, let X be a compact manifold, and let Q : C (X ) C (X) be a self-adjoint first order elliptic 9DO whose spectrum is [1, 2, 3, . . .]. Let A be a zeroth order 9DO on X whose spectrum does not contain zero in its closed convex hull. Write P n for the projector onto the span of the eigenfunctions of Q with eigenvalues in [1, ..., n]. Our main result is a second order asymptotic formula for log det P n AP n , as n . 1997 Academic Press
INTRODUCTION Let S 1 denote the circle R2?Z and let Pn denote the space of functions on S 1 spanned by [e im% : 0 m n&1]. Write P n for the orthogonal projection L 2(S 1 ) Pn . For f # L 1(S 1 ) let f n denote the n th Fourier coefficient of f ; f n =
|
f (%) e &in%
S1
d% . 2?
For a function f on S 1 let M f denote the operator multiplication by f. The Strong Szego Limit Theorem. If the function f : S 1 C has a logarithm satisfying @ f m | 2 < , : |m| |log m#Z
* Supported by NSF Grant DMS 890771. Supported by NSF Grant DMS 9506057.
496 0022-123697 25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved.
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SPECTRAL ASYMPTOTICS
then log det P nM f P n =n
|
2?
log f (%) 0
+o(1)
d% @ + : m log fm @ log f &m 2? m=1
as
modulo 2?i
n .
(0.1)
This result is due to G. Szego, see [GS]. Throughout this paper, X is a smooth, compact, closed, d dimensional manifold, Q: C (X ) C (X) is a self-adjoint first order elliptic 9DO with spectrum [1, 2, 3, . . .]. Write q(x, !) for the principal symbol of Q, and consider the bicharacteristic vector field d
3= : i=1
\
q q & . ! i x i x i ! i
+
Since the spectrum of Q is [1, 2, 3, . . .], the trajectories of this vector field are all periodic with period 2?, see [DG]. We assume that the trajectories are all simply periodic with period 2?. Such an operator Q is called an operator of Zoll type. Given a point (x, !) # T *(X ), let #(s; x, !), 0 s 2? be the trajectory of 3 with #(0 ; x, !)=(x, !). Write ? k for the projector onto the eigenspace of Q with eigenvalue k, for convenience set ? k =0 if k 0, and write P n = k n ? k . A is a zeroth order 9DO on X. Write a(x, !) for the principal symbol of A. In general for a pseudodifferential operator B, write _ 0(B)(x, !) for the principal symbol of B. This is a function on T*(X )"0. We decompose A into its Fourier coefficients as
A= :
An ,
n=&
where A n =: ? n+k A? k k
=
1 2?
|
2?
e inte &itQAe itQ dt
0
By Egorov [E], the operator A n is a zeroth order 9DO. Moreover, its symbol is given by _(A n )(x, !)=
1 2?
|
2?
_(A)(#(t ; x, !)) e int dt.
0
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(0.2)
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GUILLEMIN AND OKIKIOLU
Theorem 0.1. (The analogue of the strong Szego limit theorem for Zoll operators). Suppose that the closed convex hull of the spectrum of A does not contain the origin, and that the principal symbol of Q satisfies q(x, &!)= q(x, !). Then the following asymptotic expansion holds: trace log(P nAP n )&trace P n log AP n =
n d&1d : j 2 d+1? d j=1
||
_ 0((log A) j )(x, !) _ 0((log A) & j )(x, !) d! dx
q(x, !)<1
+o(n d&1 ),
(0.3)
Remark. If we drop the assumption that q(x, &!)=q(x, !), then a more complicated term appears on the right hand side of (0.2). The interested reader can obtain this formula by following our proof without using (1.3). We now write down explicitly the asymptotics of trace P n log AP n and hence give an explicit formula for log det(P n AP n )=trace log(P n AP n ). We first introduce the GuilleminWodzicki residue for pseudodifferential operators. Let B be a pseudodifferential operator on X of integral order ;, with symbol expansion in local coordinates _(B)(x, !)tb ;(x, !)+b ;&1(x, !)+ } } } where b j (x, !) is homogeneous in ! of degree j. Then 1 (2?) d
\|
+
b &d (x, !) d! dx |!| =1
is a well defined density on M. (It does not depend on the coordinates in which it is computed.) Define the residue of B Res(B)=
1 (2?) d
| | M
b &d (x, !) d! dx. |!| =1
Now the principal symbol q(x, !) of Q is positive, and Res(Q &dA)= =
1 (2?) d
| |
d (2?) d
||
M
(q(x, !)) &d a(x, !) d! dx |!| =1
a(x, !) d! dx,
(0.4)
[(x, !) # T *X : q(x, !)<1]
which clearly only depends on the principal symbols of A and Q. We remark that the integral in (0.3) is (n d&12) Res(Q &d (log A) j (log A) & j ).
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499
Lemma 0.2. trace ? n A? n t
c k n k,
:
c k =Res(Q &1&kA).
&
For the proof, see the appendix. From Lemma 2, we see that n
trace P n AP n = : trace ? j A? j j=1 n
n
=c d&1 : j d&1 +c d&2 : j d&2 +O(n d&2 ) j=1
j=1
n d&1 n d n d&1 + Res(Q 1&dA)+O(n d&2 ). = Res(Q &dA)+ d 2 d&1
\
+
Combining this with Theorem 0.1, we get immediately: Theorem 0.1$. If the closed convex hull of the spectrum of A does not contain the origin, and q(x, !)=q(x, &!), then the following asymptotic expansion holds: trace log(P n AP n )=
\
n d&1 n d n d&1 + Res(Q 1&d log A) Res(Q &1 log A)+ d 2 d&1
+
+
n d&1 : j Res(Q &d (log A) j (log A) & j )+o(n d&1 ). 2 j=1 (0.5)
We calculate the residues on the right explicitly in the following special case. Let X be a Zoll manifold, that is a compact, closed, d dimensional Riemannian manifold all of whose geodesics are closed and simple with length 2?. Write 2 for the LaplaceBeltrami operator on X. There exists a constant : such that the spectrum of - &2 is concentrated in bands around the points k+(:4), k # [1, 2, . . .], see [DG]. Indeed, there exists V, a 9DO of order &1, such that Q=- &2&(:4)+V is Zoll and [V, 2]=0, see [CV]. Let P n be defined as a spectral projector for Q as above, so P n is the projector onto the span of the eigenfunctions of - &2 with eigenvalues in the first n bands. Theorem 0.3. Let X be a Zoll manifold, and suppose that f is a smooth complex valued function on X such that the convex hull of the image of f does not contain zero, and let M f be the operator, multiplication by f. Then the following asymptotic formula holds:
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GUILLEMIN AND OKIKIOLU
trace log(P n M f P n )=
Vd 1 : n d +dn d&1 + (2?) d 2 4
\
+
\ ++ |
n d&1 8(2?) d+1
||
log f (x) - g(x) dx X
(log f (x)&log f ( y)) 2 sin 2(dist(x, y)2) X_X
_#(x, y) - g(x) - g( y) dx dy+o(n d&1 ),
(0.6)
where V d is the volume of the unit ball in R d, g(x) is the determinant of the metric tensor, and #(x, y) is defined as follows. For x # X fixed, the exponential map on T x X can be used to define polar coordinates (r, %) on a dense open subset of X. Here, 0 r
: jh j h & j = j=1
1 8(2?) 2
2?
| | 0
2?
0
(h(x)&h( y)) 2 dx dy. sin 2 12 (x& y)
(0.7)
For the d dimensional sphere, #(x, y)=sin 1&d (dist(x, y)). For the 2 and 3 dimensional spheres formula (0.5) was obtained in [Ok, 1] and [Ok, 2]. For the d dimensional sphere, (0.6) was conjectured in [Ok, 2]. Some other analogues of the strong Szego limit theorem are listed in Section 2.
1. PROOFS First we calculate the asymptotics of the moments. Lemma 1.1.
If q(x, !)=q(x, &!), then
trace(P n AP n ) s &trace P n A sP n =
&n d&1 s&1 sj : : Res(Q &d (A r ) j (A s&r ) & j )+o(n d&1 ). 2 r(s&r) r=1 j=1
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(1.1)
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SPECTRAL ASYMPTOTICS
Later, by making a Taylor expansion of the logarithm at the identity, we use (1.1) to prove Theorem 0.1. At the end of this section, we prove Theorem 0.3 by calculating residues. Proof of Lemma 1.1. Start by assuming that A has only finitely many Fourier terms, i.e., for some M, A k =0 for |k| >M. For any l and any list of real numbers k 1 , ..., k l , define m (k 1 , ..., k l )=min[0, k 1 , k 1 +k 2 , ..., k 1 + } } } +k l ]. * Using the fact that A k P n =P n+k A k , we get (P n AP n ) s = :
Pn Ak 1 Pn } } } Ak s Pn
k 1 , ..., k s
= :
P n P n+k 1 } } } P n+k 1 + } } } +k s A k 1 } } } A k s
k 1 , ..., k s
= :
P n+m
k 1 , ..., k s
= :
*
(k 1 , ..., k s )
Ak 1 } } } A k s
Pn Ak 1 } } } A k s & :
k 1 , ..., k s
:
? n+ j A k 1 } } } A k s
k 1 , ..., k s m (k 1 , ..., k s )< j 0 *
=P n A s & :
:
? n+ j A k 1 } } } A k s .
(1.2)
k 1 , ..., k s m (k 1 , ..., k s )< j 0 *
We remark that since A has only finitely many Fourier terms, all the above sums are finite. Now since ? n A k 1 } } } A k s = ? n A k 1 } } } A k s ? n & k 1 & } } } & k s , we see that trace ? n A k 1 } } } A k s =0 unless k 1 + } } } +k s =0. From Lemma 0.2, we see that trace ? n A k 1 } } } A k s =n d&1 Res(Q &dA k 1 } } } A k s )+o(n d&1 ) as n , hence trace(P n AP n ) s &trace P n A sP n =& :
:
trace ? n+ j A k 1 } } } A k s
k 1 , ..., k s m*(k 1 , ..., k s )< j 0
=n d&1
: k 1 + } } } +k s =0
m (k 1 , ..., k s ) Res(Q &dA k 1 } } } A k s )+o(n d&1 ). *
Now by (0.4), Res(Q &dA k 1 } } } A k s ) is independent of permutations of [1, ..., s]. Write Cr for the cyclic group generated by (1, ..., r).
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GUILLEMIN AND OKIKIOLU
(HuntDyson formula)
Lemma 1.2.
: (m (k { 1 , ..., k { r )&m (k { 1 , ..., k { r&1 ))=(k 1 + } } } +k r ) & , * *
{ # Cr
where %t =
%
%<0, % 0.
{0,
We remark that :
A k 1 } } } A k r =(A r ) j .
k 1 + } } } +k r = j
From (1.2), we have trace(P n AP n ) s &traceP n A sP n =n d&1
: k 1 + } } } +k s
=n d&1
m (k 1 , ..., k s ) Res(Q &dA k 1 } } } A k s )+o(n d&1 ) * =0
: k 1 + } } } +k s =0 1 r s&1
(m (k 1 , ..., k r )&m (k 1 , ..., k r&1 )) * *
_Res(Q &dA k 1 } } } A k s )+o(n d&1 ) =n d&1
1 : (m (k { , ..., k { r )&m (k { 1 , ..., k { r&s )) * r { # Cr * 1 k 1 + } } } +k s =0 :
1 r s&1
_Res(Q &dA k 1 } } } A k s )+o(n d&1 ) =n d&1
1 (k 1 + } } } +k r ) & Res(Q &dA k 1 } } } A k s )+o(n d&1 ) r k 1 + } } } +k s =0 :
1 r s&1 s&1
=n d&1 : r=1
:
:
j=1 k 1 + } } } +k r =&j l 1 + } } } +l s&r = j
&j Res(Q &dA k 1 } } } A k r A l 1 } } } A l s&r ) r
+o(n d&1 ) s&1
=&n d&1 : r=1
: j=1
j Res(Q &d (A r ) & j (A s&r ) j )+o(n d&1 ). r
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SPECTRAL ASYMPTOTICS
To complete the proof of (1.1) for the case when A has finite Fourier expansion, we must show that s&1
: r=1
1 Res(Q &d (A r ) & j (A s&r ) j ) r =
1 s&1 s : Res(Q &d (A r ) j (A s&r ) & j ) 2 r=1 r(s&r)
(1.3)
We first remark that for zero order operators A and B, Res(Q &dA & j B j )=Res(Q &dA j B & j ). Indeed, we have _ 0(A & j )(x, !)= =
1 2?
|
1 2?
|
2?
_ 0(A)(#(t ; x, !)) e &ijt dt
0 2?
_ 0(A)(#(t; x, &!)) e ijt dt=_ 0(A j )(x, &!)
0
and making the change of variable (x, !) (x, &!), Res(Q &dA & j B j )= =
d (2?) n
||
_ 0(A & j )(x, !) _ 0(B j )(x, !) d! dx
d (2?) n
||
_ 0(A j )(x, &!) _ 0(B & j )(x, &!) d! dx
q(x, !)<1
q(x, !)<1
=Res(Q &dA j B & j ). Here we have used the fact that q(x, !)=q(x, &!). Now s&1
: r=1
s&1 1 1 Res(Q &d (A r ) & j (A s&r ) j )= : Res(Q &d (A r ) j (A s&r ) & j ) r s&r r=1 s&1
=: r=1
1 Res(Q &d (A r ) j (A s&r ) & j ) r
Taking the mean of the second two expressions yields (1.3), since 1 s 1 + = . 2(s&r) 2r 2r(s&r) Now we extend (1.1) to any zero order 9DOA. We write &A& for the operator norm of A.
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GUILLEMIN AND OKIKIOLU
Lemma 1.3. If A is a zero order DO, then the sequence &A j & is rapidly decreasing in j. Proof. Write Ad Q(A)=[Q, A]. If A is zero order, then so is Ad Q(A). We will show that j N &A j & &(Ad Q) n (A)&, which proves the Lemma. The case N=0 is easy to prove. For N=1, first write B=B n =P nAP n . Then [Q, B]=: [Q, B j ]=: [Q, ? k+ j B? k ]=: j? j+k B? k =: jB j . j
j, k
j, k
j
Thus & jB j & &[Q, B]&=&[Q, P n AP n ]&=&P n[Q, A] P n & &[Q, A]&. The first inequality follows from the fact that the terms jB j are the Fourier coefficients of [Q, A]. Now &A j &=lim n &B (n) j &, and so we are done. A similar argument works for bigger values of N, after observing by induction that (Ad Q) N (B)= j j NB j . Now define &A& =&A&+sup n (1&d )2 &P n A(I&P n )& 2 . * n &AB& &A& &B&+&A& &B& . * * * (b) &A & r&A& r&1 &A& . * * (c) &AB& &A& &B& * * * (d)
Lemma 1.4.
(a)
r
: j |Res(Q &dA & j B j )| &A& &B& . * * j=1 Proof. &P n AB(I&P n )& 2 &P n AP n B(I&P n )& 2 +&P n A(I&P n ) B(I&P n )& 2 &P n A& &P n B(I&P n )& 2 +&P n A(I&P n )& 2 &B(I&P n )& &A& &P n B(I&P n )& 2 +&P n A(I&P n )& 2 &B&.
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SPECTRAL ASYMPTOTICS
From this, (a), (b), (c) follow easily. For (d), &P n A(I&P n )& 22 =trace P n A(I&P n ) A*P n
= :
trace P n A & j (I&P n ) A j* P n
j=&
= :
trace ? k A & j A j* ? k
:
j=& n& j
tn d&1
:
j Res(Q &dA & j A *). j
j=&
To get the final line, use Lemmas 0.2, 1.3 and the dominated convergence theorem. (d) then follows by Cauchy-Schwarz. Write t 0n(s ; A)=
1 trace(&(P n AP n ) s +P n A sP n ) n d&1
t 1n(s ; A)=
1 s&1 j : : Res(Q &d (A r ) & j (A s&r ) j ). 2 r=1 j=1 r(s&r)
For :=0, 1, t :n(s ; A) s 2 &A& s&2 &A& 2 . * If &A& , &B&
Lemma 1.5. (b)
(a)
|t :n(s ; A)&t :n(s ; B)| sM s&1 &A&B& . * Proof of Lemma 1.5. (a) |t 0n(s ; A)| =
1 n
d&1
1 n d&1 1 n
d&1
}
s&1
trace : (P n A) r (I&P n ) A s&rP n r=1
}
s&1
: &(P n A) r (I&P n )& 2 &(I&P n ) A s&rP n & 2 r=1 s&1
: &(P n A) r&1& &P n A(I&P n )& 2 &(I&P n ) A s&rP n & 2 r=1
s&1
: (s&r) &A s&2 & &A& 2 , * r=1
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GUILLEMIN AND OKIKIOLU
by Lemma 1.4(b), and |t 1n(s ; A)| =
}
1 s&1 j : : Res(Q &dA r& j A s&r ) j 2 r=1 j=1 r(s&r)
}
1 s&1 1 : &A r& &A s&r& * * 2 r=1 r(s&r)
s &A s&2 & &A& 2 . * 2 (b) |t 0n(s ; A)&t 0n(s ; B)| =
1 n d&1
r=1
s&1
1 n
}
s&1
trace : ((P n A) r (I&P n ) A s&rP n &(P n B) r (I&P n ) B s&r P n )
d&1
: |trace(P n A) r (I&P n )(A s&r &B s&r ) P n r=1
+trace((P n A) r &(P n B) r )(I&P n ) B s&rP n |
s&1
1 n d&1
: (&(P n A) r (I&P n )& 2 &(I&P n )(A s&r &B s&r ) P n & 2 r=1
+&((P n A) r &(P n B) r )(I&P n )& 2 &(I&P n ) B s&rP n & 2 )
s&1
1 n
d&1
: (&(P n A) r (I&P n )& 2 &(I&P n )(A s&r &B s&r ) P n & 2 r=1
+&((P n A) r&1 + } } } +(P n B) r&1 ) P n(A&B)(I&P n )& 2 _&(I&P) n B s&rP n & 2 ) M r &A s&r &B s&r& +rM s&1 &A&B& sM s&1 &A&B& . * * * Also, |t 1n(s ; A)&t 1n(s; B)| =
1 s&1 j : : (Res(Q &d (A r &B r ) & j (A s&r ) j ) 2 r=1 j=1 r(s&r) +Res(Q &d (B r ) & j )(A s&r &B s&r ) j )
1 s&1 1 : (&A r &B r& &A s&r& +&B r& &A s&r &B s&r& ) * * * * 2 r=1 r(s&r)
s M s&1 &A&B& . * 2
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}
507
SPECTRAL ASYMPTOTICS
Lemma 1.1 states that for zero order pseudodifferential operators A, t 0n(s ; A)&st 1n(s; A)=o(n d&1 )
as
n .
(1.4)
We proved that (1.4) holds when A is replaced by A (M) = |k|
If A is a zero order pseudodifferential operator on X with
trace log(P n(I&A) P n )&trace P n log(I&A) P n =
n d&1 : j Res(Q &d (log(I&A)) j (log(I&A)) & j )+o(n d&1 ). 2 j=1 Let t 0 and t 1 be defined as above. Now
Proof of Lemma 1.6.
trace log P n(I&A) P n &trace P n log(I&A) P n
=: s=2
1 (&trace(P n AP n ) s +trace P n a sP n ) 2
=n d&1 : s=2
1 0 t (s ; A), s n
and
: j Res(Q &d (log(I&A)) j (log(I&A)) & j ) j=1
=:
:
:
r=1
u=1
j=1
s&1
:
:
=: s=2
r=1 j=1
j Res(Q &d (A r ) j (A u ) & j ) ru j Res(Q &d (A r ) j (A s&r ) & j ) r(s&r)
= : t 1n(s ; A). s=2
The sums here converge absolutely, since
: j |Res(Q &d (A r ) j (A u ) & j )| C &A r& &A u& C &A& r+u&2 &A& 2 . * * * j=1
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GUILLEMIN AND OKIKIOLU
Hence
} trace log P (I&A) P &trace P log(1&A) P n
n
n
n
n d&1 : j Res(Q &d (log(I&A)) j (log(I&A)) & j ) & 2 j=1
n d&1 : s=2
}
}
1 0 t (s ; A)&t 1n(s ; A) . s n
}
By Lemma 1.5(a), (1.4) and the dominated convergence theorem we see that this final expression is o(n d&1 ) as n , which completes the proof of Lemma 1.6 and hence Theorem 0.1. Proof of Theorem 0.3. Working in normal coordinates, x=x 1, ..., x d ) around a fixed point p # X, &2 has symbol |!| 2 +O(|!| |x| )+O(|!| 2 |x| 2 ). Using the symbolic calculus, we find that if Q=- &2&(:4) I, then Q &d has principal symbol |!| &d at x=0, while Q 1&d has principal symbol |!| 1&d and the second term in the symbol expansion is (:(d&1)4) |!| &d at x=0. Hence for a smooth function h on X, Res(Q &dM h )=
vol(S d&1 ) (2?) d
|
h(x) - g(x) dx,
X
and Res(Q 1&dM h )=
:(d&1) vol(S d&1 ) 4 (2?) d
| h(x) - g(x) dx.
Combining this with (0.5) gives the first term on the right hand side of (0.6). Let B be a zero order 9DO on X with principal symbol b(x, !)=b(x). We have Res(Q &dB j B & j )=
d (2?) d
|
_(B j )(x, !) _(B & j )(x, !) d! dx,
q(x, !)<1
and by (0.2) and (0.7),
: j Res(Q &dB j B & j ) j=1
=
d 8(2?) d+2
|
q(x, !)<1
2?
2?
0
0
| |
(b(#(t ; x, !))&b(#(s ; x, !))) 2 ds dt d! dx. sin 2((t&s)2) (1.5)
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Set (x$, %)=#(x, !, s), r=t&s. Then ds dt d! dx=ds dr d% dx$, and (1.5) is equal to d 8(2?) d+2 =
2?
|
q(x$, %)<1
1 4(2?) d+1
| | 0
| | X
2? 0
|%| =1
(b(#(r; x$, %))&b(x$)) 2 dr ds d% dx$ sin 2(r2)
|
? 0
(b(#(r ; x$, %))&b(x$)) 2 dr d% - g(x$) dx$. sin 2(r2)
From this, we get the second term in the right in (0.6).
2. CONCLUDING REMARKS We conclude with a few remarks about multidimensional strong Szego limit theorems which are not specifically associated with a Zoll operator on a compact manifold (but are, for the most part, associated with other Zoll type phenomena): 1. Let A be a zeroth order pseudodifferential operator on R d whose Weyl symbol is polyhomogeneous with respect to the homotheties, (x, !) (* 12x, * 12!), and let P n be projection onto the subspace of L 2(R d ) spanned by the Hermite functions of degree n. Then trace (P n AP n ) s admits a two-tiered asymptotic expansion of the form (1.1) in which Q is the harmonic oscillator. (See [JZ].) 2. Let B d be the closed unit ball in C d and H 2 the space of L 2 holomorphic functions on B d. Given f # C (B d ) let T f be the contraction to H 2 of the operator multiplication by f, and let P n be the projection of H 2 onto the space spanned by the polynomials, : a : z :,
z=(z 1 , ..., z d ).
|:| n
Then trace (P n T f P n ) s admits a two-tiered asymptotic expansion of the form (1.1) involving ``Toeplitz'' analogues of the residues (0.4) (cf. [Gu, 2]) in particular the replacement for Q in (0.4) is the operator, i z r z r .) 3. For d=1 the example above yields the classical Szego theorem (0.1). Both of the previous examples are special cases of a Szego theorem for multidimensional Toeplitz operators of the type discussed in [BG]. (The first example can be translated into a theorem of this type by means of the standard isomorphism between L 2(R d ) and the Bargmann space of holomorphic functions on C d which are L 2 summable with respect to the measure, exp( & |z| 2 ) dz dz.) We will discuss these and other Toeplitz analogues of the theorems in this paper in a future article.
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4. There is an analogue of the Szego theorem (0.1), where the circle S 1 is replaced by the interval [ &1, 1], and P n is the orthogonal projection onto the polynomials of degree at most n. Here, changing the measure on [ &1, 1] gives rise to different projections P n . Special cases are the Szego theorems for ultraspherical polynomials. (See [Hi].) In [Ok, 2], still working on the interval, P n is the spectral projection for a general Sturm Liouville operator with Dirichlet or Neumann boundary conditions. In [J1] an analogue of (0.1) is proved where S 1 is replaced by a general smooth Jordan curve in the complex plane, and P n is the orthogonal projection onto the polynomials in z of degree at most n, with respect to arc length. 5. If a compact Lie group, K, acts on X and this action preserves Q, there are various equivariant versions of Theorem 0.1. For instance let X=S 2 and let S 1 act on S 2 by rotation about the z-axis. Let A be an S 1 invariant zeroth order pseudodifferential operator and let 1
1 : L 2(S 2 ) L 2(S 2 ) S 1
be orthogonal projection. (Here, L 2(S 2 ) S denotes the S 1 invariant functions in L 2(S 2 ).) Then there is a strong Szego theorem for trace (1P nAP N 1 ) s (which is basically just the Szego theorem for Legendre polynomials first proved in [Hi]). More generally for arbitrary K and X, suppose the action of K on X has the property ( V ) below: For x # X let N x be the co-normal space to K } x at X and let K x be the stabilizer group of x in K. (Note that K x acts linearly on N x .) For all x # X and all ! # N x &0, the stabilizer group of ! in K x is finite. (V) This property implies that the projection operator 1: L 2(X) L 2(X) K is a Fourier integral operator, and one can deduce from this fact a two-tiered Szego asymptotics for trace (1P n AP n 1 ) s analogous to (1.1). 6. Several years ago one of us proved that for any positive self-adjoint first order elliptic pseudodifferential operator, Q: C (X) C (X), one has a weak Szego limit theorem trace(P * AP * ) k =(2?) &d * d
|
a(x, !) k dx d!+o(* d ),
(2.1)
q(x, !)<1
P * being projection onto the eigenfunctions of Q with eigenvalue <*. (See [Gu, 1].) Moreover Hormander ([Ho, 1], Section 29.1) showed that the
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o(* d ) on the right can be replaced by O(* d&12 ), and recently Laptev and Safarov [LS] have shown that, in fact, one can replace o(* d ) by O(* d&1 ). Moreover, it is not hard to show that for k=1 in (2.1), the error term on the right is o(* d&1 ) providing the set of periodic trajectories in T*X&O of the bicharacteristic flow associated with q(x, !) is of measure zero (in other words providing Q is non-Zoll in a rather strong sense). These results suggest that there may be two-tiered Szego limit theorems for other operators besides Zoll operators; and, in fact, there are, as the following two examples show. 7. There is an analogue of Lemma 1.1 where X is replaced by R d, A is a pseudo-differential operator on R d of sufficiently negative order, and for any fixed domain 0 in R d, P * is multiplication on the Fourier transform side by the characteristic function of *0. (See [W]). 8. Similar to the case described in 7, there is an analogue of Theorem 0.3 where X is replaced by the flat d-dimensional torus, and for any fixed smooth domain 0 in R d satisfying certain weak conditions, P * is truncation operator on the Fourier transform side to the finite Fourier series supported on *0. (See [Do], [L], [Ok, 1].) Kurt Johansson [Jo, 2] noticed recently that there is a remarkable connection between the asymptotics (0.1) and the random behavior of n_n unitary matrices as n goes to infinity. 1 It would be interesting to know whether the multidimensional Szego theorems described here have similar implications in random matrix theory.
APPENDIX We will give here details of the proof of Lemma 2. We begin by observing that if the bicharacteristic flow on T*X&0 associated with q(x, !) is simply periodic of period 2? then &
trace(exp itQ) At :
c k(A) / k(t)
(1)
k=d&1
where / k(t)= : n ke int.
(2)
n>0 1 He used this observation to settle a conjecture of DiaconisShahshahani which says that the random variables, ( j) &12 trace M j, tend, in a very strong sense, to Gaussian i.i.d.s as n goes to infinity.
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(Indeed, by [DG], Section 4, the left hand side of (1) is a classical co-normal distribution on the circle with wave front set at t=0, {>0, and every such distribution admits an asymptotic expansion of the form above.) Rewriting the left hand side in the form : e itn trace ? n A? n and comparing Fourier coefficients one deduces &
trace ? n A? n t :
c k(A) n k
(3)
k=d&1
We will now show that the c k(A)'s can also be characterized as symbolic residues: Consider the zeta function ` A(z)=trace Q zA. This function has simple poles at z=&d, &d+1, &d+2, etc., and we will show that the residues at these poles are c d&1(A), c d&2(A), etc. (In other words, c k(A)=Res(Q &k&1A).) We will deduce this result from the following more general result: Let Q be any positive self-adjoint first order elliptic 9DO of order one. (i.e. not necessarily an operator of Zoll type.) By [Ho, 1], Section 4 there is an asymptotic expansion of the form &
trace(exp itQ) At :
c k(A) / k(t)
(4)
k=d&1
near t=0 where / k(t)=
|
t ke its ds,
* 0 >0.
(5)
*0
(This / k is not the same as the / k defined by (2), however; it differs from it by a function which is smooth in a neighborhood of zero; so the asymptotic expansion (4) is identical with the asymptotic expansion (1). Theorem. The c k(A)'s occurring in (4) are the residues at the points, z=&k&1, of the zeta function ` A(z)=trace Q zA.
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Proof. Let P * be projection onto the span of the eigenfunction of P of eigenvalue * i *, and let _(*)=trace P * AP * . Then @ d_ def (t)=trace(exp itQ) A = e(t). d* Let \ be a function whose Fourier transform is C , is supported in the set |t| <= and is one on the set |t| <=2. For small enough = \^
& @ d_ (t)t : c k(A) /^ k(t); d* k=d&1
so by taking inverse Fourier transforms: & d_ t : c k(A) * k d* k=d&1
\V
(6)
for *> >0. Consider now the zeta function above, ` A(z)=
|
*z
*0
d_ d*, d*
(7)
where 0<* 0
|
*z
*0
\
d_ d_ &\ V d* d* d*
+
(8)
is holomorphic in z in the whole z plane. In other words
|
*z
*0
d_ d* d*
differs from
|
\
*z \ V
*0
d_ d* d*
+
(9)
by a holomorphic function of z. Thus if one substitutes the expansion (6) into (9) one gets &N
` A(z)= : k=d&1
c k(A)
|
* z+k d*+ } } }
1
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GUILLEMIN AND OKIKIOLU
where
|
* z+k d*=
1
d z+k+1 1 * z+k+1 d*
}
1
1 =& z+k+1
and the `` } } } '' is a function which is holomorphic for Re z* 0 z and is one for *>* 0 . Setting / z (*)=#(*) * z and f (*)=
d _(*) d*
it suffices to show that
|
/ z (*)( f(*)&\ V f (*)) d*
&
is holomorphic in the whole z plane. Therefore, if one rewrites this expression as
|
/$ s(t)(1&\^(t)) f (t) dt
&
and notes that f (t) is a tempered distribution in t, it suffices to prove. Lemma. Proof.
K
(1&\^ ) /$ z is an entire function of z (with values in S(R).) By definition of the inverse Fourier transform /$ z (t)=(2?) &1
|
e i*t#(*) * z d*
&
=(2?) &1
i t
k
\+ |
e i*t
&
d d*
k
\ + #(*) * d* z
Since 1&\^ is zero when |t| <=2, one can show from this that (1&\^ ) /$ z is bounded in all the semi-norms d dt
s
} \ + f}
| f | r, s =sup t r
uniformly on compact subsets of the z-plane. K
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Note added in proof. After submitting this paper for publication, we noticed that this second-order asymptotic formula could be replaced by an asymptotics to all orders. For a brief description of this result, which depends very much on the results below, see the research announcement [GO].
REFERENCES [BG] [CV] [Do] [DG] [E] [GS] [Gu, 1] [Gu, 2] [GO] [Hi] [Ho, 1] [Ho, 2] [Jo, 1]
[Jo, 2] [JZ] [K] [LS] [Li] [Ok, 1] [Ok, 2]
L. Boutet de Monvel and V. Guillemin, ``The Spectral Theory of Toeplitz Operators,'' Princeton Univ. Press, Princeton, NJ, 1981. Y. Colin de Verdiere, Sur le spectre des operateurs elliptiques a bicaracteristiques toutes periodiques, Comm. Math. Helv. 54 (1979), 508522. R. Doktorski@$ , Generalization of the Szego limit theorem to the multidimensional case, Sibirsk. Math. Zh. 25 (5) (1984), 2029. J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1974), 3979. Ju. V. Egorov, The canonical transformations of pseudodifferential operators, Uspehi Mat. Nauk. 24 (1969), 235236. U. Grenader and G. Szego, ``Toeplitz Forms and Their Applications,'' Calif. Mono. in Math. Sci., Univ. of California Press, Berkeley and Los Angeles, 1958. V. Guillemin, ``Some Classical Theorems in Spectral Theory Revisited,'' pp. 219259, Princeton Univ. Press, Princeton, NJ, 1981. V. Guillemin, Residue traces for certain algebras of Fourier integral operators, J. Funct. Anal. 115 (1993), 391417. V. Guillemin and K. Okikiolu, Szego theorems for Zoll operators, Math. Res. Lett. 3 (1996), 14. I. Hirschman, Jr., The strong Szego limit theorem for Toeplitz determinants, Amer. J. Math. 88 (1966), 577614. L. Hormander, ``The Analysis of Linear Partial Differential Operators,'' SpringerVerlag, BerlinHeidelbergNew York, 1985. L. Hormander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 191218. K. Johansson, ``On Szego's Asymptotic Formula for Toeplitz Determinants and Generalizations,'' Ph.D. diss., UUDM Report 15, Uppsala Univ., Dept. of Math., Uppsala, Sweden, 1987. K. Johansson, On random matrices from the compact classical groups, preprint, Dept. of Math., Royal Institute of Technology, S-100 44, Stockholm, Sweden. A. Janssen and S. Zelditch, Szego limit theorems for the harmonic oscillator, Trans. Amer. Math. Soc. 80 (2) (1983), 563587. M. Kac, Toeplitz matrices, translation kernels and a related problem in probability theory, Duke Math. J. 21 (1954), 501509. A. Laptev and Yu Safarov, Szego type limit theorems, J. Funct. Anal. 138 (1996), 544559. I. Linik, A multidimensional analog of a limit theorem of G. Szego, Math. USSR Izv. 9 (1975), 13231332. K. Okikiolu, ``The Analogue of the Strong Szego Limit Theorem on the Torus and the 3-Sphere,'' Ph.D. diss., Dept of Math., UCLA, Los Angeles, CA, 1991. K. Okikiolu, The analogue of the strong Szego limit theorem on the 2 and 3 dimensional spheres, J. Amer. Math. Soc., to appear.
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[Ok, 3] K. Okikiolu, An analogue of the strong Szego limit theorem for a SturmLiouville operator on the interval, preprint. [U] A. Uribe, A symbol calculus for a class of pseudodifferential operators on S n and band asymptotics, J. Funct. Anal. 59 (1984), 535556. [W] H. Widom, ``Asymptotic Expansions for Pseudodifferential Operators in Bounded Domains,'' Lect. Notes in Math., Vol. 1152, Springer-Verlag, BerlinNew York, 1985.
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Spectral Asymptotics of Toeplitz Operators on Zoll Manifolds V. Guillemin* and K. Okikiolu Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received June 20, 1996; revised August 30, 1996; accepted September 2, 1996
We prove a generalization of the Strong Szego Limit Theorem for Zoll type operators on smooth compact closed manifolds. More precisely, let X be a compact manifold, and let Q : C (X ) C (X) be a self-adjoint first order elliptic 9DO whose spectrum is [1, 2, 3, . . .]. Let A be a zeroth order 9DO on X whose spectrum does not contain zero in its closed convex hull. Write P n for the projector onto the span of the eigenfunctions of Q with eigenvalues in [1, ..., n]. Our main result is a second order asymptotic formula for log det P n AP n , as n . 1997 Academic Press
INTRODUCTION Let S 1 denote the circle R2?Z and let Pn denote the space of functions on S 1 spanned by [e im% : 0 m n&1]. Write P n for the orthogonal projection L 2(S 1 ) Pn . For f # L 1(S 1 ) let f n denote the n th Fourier coefficient of f ; f n =
|
f (%) e &in%
S1
d% . 2?
For a function f on S 1 let M f denote the operator multiplication by f. The Strong Szego Limit Theorem. If the function f : S 1 C has a logarithm satisfying @ f m | 2 < , : |m| |log m#Z
* Supported by NSF Grant DMS 890771. Supported by NSF Grant DMS 9506057.
496 0022-123697 25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved.
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SPECTRAL ASYMPTOTICS
then log det P nM f P n =n
|
2?
log f (%) 0
+o(1)
d% @ + : m log fm @ log f &m 2? m=1
as
modulo 2?i
n .
(0.1)
This result is due to G. Szego, see [GS]. Throughout this paper, X is a smooth, compact, closed, d dimensional manifold, Q: C (X ) C (X) is a self-adjoint first order elliptic 9DO with spectrum [1, 2, 3, . . .]. Write q(x, !) for the principal symbol of Q, and consider the bicharacteristic vector field d
3= : i=1
\
q q & . ! i x i x i ! i
+
Since the spectrum of Q is [1, 2, 3, . . .], the trajectories of this vector field are all periodic with period 2?, see [DG]. We assume that the trajectories are all simply periodic with period 2?. Such an operator Q is called an operator of Zoll type. Given a point (x, !) # T *(X ), let #(s; x, !), 0 s 2? be the trajectory of 3 with #(0 ; x, !)=(x, !). Write ? k for the projector onto the eigenspace of Q with eigenvalue k, for convenience set ? k =0 if k 0, and write P n = k n ? k . A is a zeroth order 9DO on X. Write a(x, !) for the principal symbol of A. In general for a pseudodifferential operator B, write _ 0(B)(x, !) for the principal symbol of B. This is a function on T*(X )"0. We decompose A into its Fourier coefficients as
A= :
An ,
n=&
where A n =: ? n+k A? k k
=
1 2?
|
2?
e inte &itQAe itQ dt
0
By Egorov [E], the operator A n is a zeroth order 9DO. Moreover, its symbol is given by _(A n )(x, !)=
1 2?
|
2?
_(A)(#(t ; x, !)) e int dt.
0
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(0.2)
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GUILLEMIN AND OKIKIOLU
Theorem 0.1. (The analogue of the strong Szego limit theorem for Zoll operators). Suppose that the closed convex hull of the spectrum of A does not contain the origin, and that the principal symbol of Q satisfies q(x, &!)= q(x, !). Then the following asymptotic expansion holds: trace log(P nAP n )&trace P n log AP n =
n d&1d : j 2 d+1? d j=1
||
_ 0((log A) j )(x, !) _ 0((log A) & j )(x, !) d! dx
q(x, !)<1
+o(n d&1 ),
(0.3)
Remark. If we drop the assumption that q(x, &!)=q(x, !), then a more complicated term appears on the right hand side of (0.2). The interested reader can obtain this formula by following our proof without using (1.3). We now write down explicitly the asymptotics of trace P n log AP n and hence give an explicit formula for log det(P n AP n )=trace log(P n AP n ). We first introduce the GuilleminWodzicki residue for pseudodifferential operators. Let B be a pseudodifferential operator on X of integral order ;, with symbol expansion in local coordinates _(B)(x, !)tb ;(x, !)+b ;&1(x, !)+ } } } where b j (x, !) is homogeneous in ! of degree j. Then 1 (2?) d
\|
+
b &d (x, !) d! dx |!| =1
is a well defined density on M. (It does not depend on the coordinates in which it is computed.) Define the residue of B Res(B)=
1 (2?) d
| | M
b &d (x, !) d! dx. |!| =1
Now the principal symbol q(x, !) of Q is positive, and Res(Q &dA)= =
1 (2?) d
| |
d (2?) d
||
M
(q(x, !)) &d a(x, !) d! dx |!| =1
a(x, !) d! dx,
(0.4)
[(x, !) # T *X : q(x, !)<1]
which clearly only depends on the principal symbols of A and Q. We remark that the integral in (0.3) is (n d&12) Res(Q &d (log A) j (log A) & j ).
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499
Lemma 0.2. trace ? n A? n t
c k n k,
:
c k =Res(Q &1&kA).
&
For the proof, see the appendix. From Lemma 2, we see that n
trace P n AP n = : trace ? j A? j j=1 n
n
=c d&1 : j d&1 +c d&2 : j d&2 +O(n d&2 ) j=1
j=1
n d&1 n d n d&1 + Res(Q 1&dA)+O(n d&2 ). = Res(Q &dA)+ d 2 d&1
\
+
Combining this with Theorem 0.1, we get immediately: Theorem 0.1$. If the closed convex hull of the spectrum of A does not contain the origin, and q(x, !)=q(x, &!), then the following asymptotic expansion holds: trace log(P n AP n )=
\
n d&1 n d n d&1 + Res(Q 1&d log A) Res(Q &1 log A)+ d 2 d&1
+
+
n d&1 : j Res(Q &d (log A) j (log A) & j )+o(n d&1 ). 2 j=1 (0.5)
We calculate the residues on the right explicitly in the following special case. Let X be a Zoll manifold, that is a compact, closed, d dimensional Riemannian manifold all of whose geodesics are closed and simple with length 2?. Write 2 for the LaplaceBeltrami operator on X. There exists a constant : such that the spectrum of - &2 is concentrated in bands around the points k+(:4), k # [1, 2, . . .], see [DG]. Indeed, there exists V, a 9DO of order &1, such that Q=- &2&(:4)+V is Zoll and [V, 2]=0, see [CV]. Let P n be defined as a spectral projector for Q as above, so P n is the projector onto the span of the eigenfunctions of - &2 with eigenvalues in the first n bands. Theorem 0.3. Let X be a Zoll manifold, and suppose that f is a smooth complex valued function on X such that the convex hull of the image of f does not contain zero, and let M f be the operator, multiplication by f. Then the following asymptotic formula holds:
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trace log(P n M f P n )=
Vd 1 : n d +dn d&1 + (2?) d 2 4
\
+
\ ++ |
n d&1 8(2?) d+1
||
log f (x) - g(x) dx X
(log f (x)&log f ( y)) 2 sin 2(dist(x, y)2) X_X
_#(x, y) - g(x) - g( y) dx dy+o(n d&1 ),
(0.6)
where V d is the volume of the unit ball in R d, g(x) is the determinant of the metric tensor, and #(x, y) is defined as follows. For x # X fixed, the exponential map on T x X can be used to define polar coordinates (r, %) on a dense open subset of X. Here, 0 r
: jh j h & j = j=1
1 8(2?) 2
2?
| | 0
2?
0
(h(x)&h( y)) 2 dx dy. sin 2 12 (x& y)
(0.7)
For the d dimensional sphere, #(x, y)=sin 1&d (dist(x, y)). For the 2 and 3 dimensional spheres formula (0.5) was obtained in [Ok, 1] and [Ok, 2]. For the d dimensional sphere, (0.6) was conjectured in [Ok, 2]. Some other analogues of the strong Szego limit theorem are listed in Section 2.
1. PROOFS First we calculate the asymptotics of the moments. Lemma 1.1.
If q(x, !)=q(x, &!), then
trace(P n AP n ) s &trace P n A sP n =
&n d&1 s&1 sj : : Res(Q &d (A r ) j (A s&r ) & j )+o(n d&1 ). 2 r(s&r) r=1 j=1
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(1.1)
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SPECTRAL ASYMPTOTICS
Later, by making a Taylor expansion of the logarithm at the identity, we use (1.1) to prove Theorem 0.1. At the end of this section, we prove Theorem 0.3 by calculating residues. Proof of Lemma 1.1. Start by assuming that A has only finitely many Fourier terms, i.e., for some M, A k =0 for |k| >M. For any l and any list of real numbers k 1 , ..., k l , define m (k 1 , ..., k l )=min[0, k 1 , k 1 +k 2 , ..., k 1 + } } } +k l ]. * Using the fact that A k P n =P n+k A k , we get (P n AP n ) s = :
Pn Ak 1 Pn } } } Ak s Pn
k 1 , ..., k s
= :
P n P n+k 1 } } } P n+k 1 + } } } +k s A k 1 } } } A k s
k 1 , ..., k s
= :
P n+m
k 1 , ..., k s
= :
*
(k 1 , ..., k s )
Ak 1 } } } A k s
Pn Ak 1 } } } A k s & :
k 1 , ..., k s
:
? n+ j A k 1 } } } A k s
k 1 , ..., k s m (k 1 , ..., k s )< j 0 *
=P n A s & :
:
? n+ j A k 1 } } } A k s .
(1.2)
k 1 , ..., k s m (k 1 , ..., k s )< j 0 *
We remark that since A has only finitely many Fourier terms, all the above sums are finite. Now since ? n A k 1 } } } A k s = ? n A k 1 } } } A k s ? n & k 1 & } } } & k s , we see that trace ? n A k 1 } } } A k s =0 unless k 1 + } } } +k s =0. From Lemma 0.2, we see that trace ? n A k 1 } } } A k s =n d&1 Res(Q &dA k 1 } } } A k s )+o(n d&1 ) as n , hence trace(P n AP n ) s &trace P n A sP n =& :
:
trace ? n+ j A k 1 } } } A k s
k 1 , ..., k s m*(k 1 , ..., k s )< j 0
=n d&1
: k 1 + } } } +k s =0
m (k 1 , ..., k s ) Res(Q &dA k 1 } } } A k s )+o(n d&1 ). *
Now by (0.4), Res(Q &dA k 1 } } } A k s ) is independent of permutations of [1, ..., s]. Write Cr for the cyclic group generated by (1, ..., r).
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(HuntDyson formula)
Lemma 1.2.
: (m (k { 1 , ..., k { r )&m (k { 1 , ..., k { r&1 ))=(k 1 + } } } +k r ) & , * *
{ # Cr
where %t =
%
%<0, % 0.
{0,
We remark that :
A k 1 } } } A k r =(A r ) j .
k 1 + } } } +k r = j
From (1.2), we have trace(P n AP n ) s &traceP n A sP n =n d&1
: k 1 + } } } +k s
=n d&1
m (k 1 , ..., k s ) Res(Q &dA k 1 } } } A k s )+o(n d&1 ) * =0
: k 1 + } } } +k s =0 1 r s&1
(m (k 1 , ..., k r )&m (k 1 , ..., k r&1 )) * *
_Res(Q &dA k 1 } } } A k s )+o(n d&1 ) =n d&1
1 : (m (k { , ..., k { r )&m (k { 1 , ..., k { r&s )) * r { # Cr * 1 k 1 + } } } +k s =0 :
1 r s&1
_Res(Q &dA k 1 } } } A k s )+o(n d&1 ) =n d&1
1 (k 1 + } } } +k r ) & Res(Q &dA k 1 } } } A k s )+o(n d&1 ) r k 1 + } } } +k s =0 :
1 r s&1 s&1
=n d&1 : r=1
:
:
j=1 k 1 + } } } +k r =&j l 1 + } } } +l s&r = j
&j Res(Q &dA k 1 } } } A k r A l 1 } } } A l s&r ) r
+o(n d&1 ) s&1
=&n d&1 : r=1
: j=1
j Res(Q &d (A r ) & j (A s&r ) j )+o(n d&1 ). r
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SPECTRAL ASYMPTOTICS
To complete the proof of (1.1) for the case when A has finite Fourier expansion, we must show that s&1
: r=1
1 Res(Q &d (A r ) & j (A s&r ) j ) r =
1 s&1 s : Res(Q &d (A r ) j (A s&r ) & j ) 2 r=1 r(s&r)
(1.3)
We first remark that for zero order operators A and B, Res(Q &dA & j B j )=Res(Q &dA j B & j ). Indeed, we have _ 0(A & j )(x, !)= =
1 2?
|
1 2?
|
2?
_ 0(A)(#(t ; x, !)) e &ijt dt
0 2?
_ 0(A)(#(t; x, &!)) e ijt dt=_ 0(A j )(x, &!)
0
and making the change of variable (x, !) (x, &!), Res(Q &dA & j B j )= =
d (2?) n
||
_ 0(A & j )(x, !) _ 0(B j )(x, !) d! dx
d (2?) n
||
_ 0(A j )(x, &!) _ 0(B & j )(x, &!) d! dx
q(x, !)<1
q(x, !)<1
=Res(Q &dA j B & j ). Here we have used the fact that q(x, !)=q(x, &!). Now s&1
: r=1
s&1 1 1 Res(Q &d (A r ) & j (A s&r ) j )= : Res(Q &d (A r ) j (A s&r ) & j ) r s&r r=1 s&1
=: r=1
1 Res(Q &d (A r ) j (A s&r ) & j ) r
Taking the mean of the second two expressions yields (1.3), since 1 s 1 + = . 2(s&r) 2r 2r(s&r) Now we extend (1.1) to any zero order 9DOA. We write &A& for the operator norm of A.
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Lemma 1.3. If A is a zero order DO, then the sequence &A j & is rapidly decreasing in j. Proof. Write Ad Q(A)=[Q, A]. If A is zero order, then so is Ad Q(A). We will show that j N &A j & &(Ad Q) n (A)&, which proves the Lemma. The case N=0 is easy to prove. For N=1, first write B=B n =P nAP n . Then [Q, B]=: [Q, B j ]=: [Q, ? k+ j B? k ]=: j? j+k B? k =: jB j . j
j, k
j, k
j
Thus & jB j & &[Q, B]&=&[Q, P n AP n ]&=&P n[Q, A] P n & &[Q, A]&. The first inequality follows from the fact that the terms jB j are the Fourier coefficients of [Q, A]. Now &A j &=lim n &B (n) j &, and so we are done. A similar argument works for bigger values of N, after observing by induction that (Ad Q) N (B)= j j NB j . Now define &A& =&A&+sup n (1&d )2 &P n A(I&P n )& 2 . * n &AB& &A& &B&+&A& &B& . * * * (b) &A & r&A& r&1 &A& . * * (c) &AB& &A& &B& * * * (d)
Lemma 1.4.
(a)
r
: j |Res(Q &dA & j B j )| &A& &B& . * * j=1 Proof. &P n AB(I&P n )& 2 &P n AP n B(I&P n )& 2 +&P n A(I&P n ) B(I&P n )& 2 &P n A& &P n B(I&P n )& 2 +&P n A(I&P n )& 2 &B(I&P n )& &A& &P n B(I&P n )& 2 +&P n A(I&P n )& 2 &B&.
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SPECTRAL ASYMPTOTICS
From this, (a), (b), (c) follow easily. For (d), &P n A(I&P n )& 22 =trace P n A(I&P n ) A*P n
= :
trace P n A & j (I&P n ) A j* P n
j=&
= :
trace ? k A & j A j* ? k
:
j=& n& j
tn d&1
:
j Res(Q &dA & j A *). j
j=&
To get the final line, use Lemmas 0.2, 1.3 and the dominated convergence theorem. (d) then follows by Cauchy-Schwarz. Write t 0n(s ; A)=
1 trace(&(P n AP n ) s +P n A sP n ) n d&1
t 1n(s ; A)=
1 s&1 j : : Res(Q &d (A r ) & j (A s&r ) j ). 2 r=1 j=1 r(s&r)
For :=0, 1, t :n(s ; A) s 2 &A& s&2 &A& 2 . * If &A& , &B&
Lemma 1.5. (b)
(a)
|t :n(s ; A)&t :n(s ; B)| sM s&1 &A&B& . * Proof of Lemma 1.5. (a) |t 0n(s ; A)| =
1 n
d&1
1 n d&1 1 n
d&1
}
s&1
trace : (P n A) r (I&P n ) A s&rP n r=1
}
s&1
: &(P n A) r (I&P n )& 2 &(I&P n ) A s&rP n & 2 r=1 s&1
: &(P n A) r&1& &P n A(I&P n )& 2 &(I&P n ) A s&rP n & 2 r=1
s&1
: (s&r) &A s&2 & &A& 2 , * r=1
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GUILLEMIN AND OKIKIOLU
by Lemma 1.4(b), and |t 1n(s ; A)| =
}
1 s&1 j : : Res(Q &dA r& j A s&r ) j 2 r=1 j=1 r(s&r)
}
1 s&1 1 : &A r& &A s&r& * * 2 r=1 r(s&r)
s &A s&2 & &A& 2 . * 2 (b) |t 0n(s ; A)&t 0n(s ; B)| =
1 n d&1
r=1
s&1
1 n
}
s&1
trace : ((P n A) r (I&P n ) A s&rP n &(P n B) r (I&P n ) B s&r P n )
d&1
: |trace(P n A) r (I&P n )(A s&r &B s&r ) P n r=1
+trace((P n A) r &(P n B) r )(I&P n ) B s&rP n |
s&1
1 n d&1
: (&(P n A) r (I&P n )& 2 &(I&P n )(A s&r &B s&r ) P n & 2 r=1
+&((P n A) r &(P n B) r )(I&P n )& 2 &(I&P n ) B s&rP n & 2 )
s&1
1 n
d&1
: (&(P n A) r (I&P n )& 2 &(I&P n )(A s&r &B s&r ) P n & 2 r=1
+&((P n A) r&1 + } } } +(P n B) r&1 ) P n(A&B)(I&P n )& 2 _&(I&P) n B s&rP n & 2 ) M r &A s&r &B s&r& +rM s&1 &A&B& sM s&1 &A&B& . * * * Also, |t 1n(s ; A)&t 1n(s; B)| =
1 s&1 j : : (Res(Q &d (A r &B r ) & j (A s&r ) j ) 2 r=1 j=1 r(s&r) +Res(Q &d (B r ) & j )(A s&r &B s&r ) j )
1 s&1 1 : (&A r &B r& &A s&r& +&B r& &A s&r &B s&r& ) * * * * 2 r=1 r(s&r)
s M s&1 &A&B& . * 2
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}
507
SPECTRAL ASYMPTOTICS
Lemma 1.1 states that for zero order pseudodifferential operators A, t 0n(s ; A)&st 1n(s; A)=o(n d&1 )
as
n .
(1.4)
We proved that (1.4) holds when A is replaced by A (M) = |k|
If A is a zero order pseudodifferential operator on X with
trace log(P n(I&A) P n )&trace P n log(I&A) P n =
n d&1 : j Res(Q &d (log(I&A)) j (log(I&A)) & j )+o(n d&1 ). 2 j=1 Let t 0 and t 1 be defined as above. Now
Proof of Lemma 1.6.
trace log P n(I&A) P n &trace P n log(I&A) P n
=: s=2
1 (&trace(P n AP n ) s +trace P n a sP n ) 2
=n d&1 : s=2
1 0 t (s ; A), s n
and
: j Res(Q &d (log(I&A)) j (log(I&A)) & j ) j=1
=:
:
:
r=1
u=1
j=1
s&1
:
:
=: s=2
r=1 j=1
j Res(Q &d (A r ) j (A u ) & j ) ru j Res(Q &d (A r ) j (A s&r ) & j ) r(s&r)
= : t 1n(s ; A). s=2
The sums here converge absolutely, since
: j |Res(Q &d (A r ) j (A u ) & j )| C &A r& &A u& C &A& r+u&2 &A& 2 . * * * j=1
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GUILLEMIN AND OKIKIOLU
Hence
} trace log P (I&A) P &trace P log(1&A) P n
n
n
n
n d&1 : j Res(Q &d (log(I&A)) j (log(I&A)) & j ) & 2 j=1
n d&1 : s=2
}
}
1 0 t (s ; A)&t 1n(s ; A) . s n
}
By Lemma 1.5(a), (1.4) and the dominated convergence theorem we see that this final expression is o(n d&1 ) as n , which completes the proof of Lemma 1.6 and hence Theorem 0.1. Proof of Theorem 0.3. Working in normal coordinates, x=x 1, ..., x d ) around a fixed point p # X, &2 has symbol |!| 2 +O(|!| |x| )+O(|!| 2 |x| 2 ). Using the symbolic calculus, we find that if Q=- &2&(:4) I, then Q &d has principal symbol |!| &d at x=0, while Q 1&d has principal symbol |!| 1&d and the second term in the symbol expansion is (:(d&1)4) |!| &d at x=0. Hence for a smooth function h on X, Res(Q &dM h )=
vol(S d&1 ) (2?) d
|
h(x) - g(x) dx,
X
and Res(Q 1&dM h )=
:(d&1) vol(S d&1 ) 4 (2?) d
| h(x) - g(x) dx.
Combining this with (0.5) gives the first term on the right hand side of (0.6). Let B be a zero order 9DO on X with principal symbol b(x, !)=b(x). We have Res(Q &dB j B & j )=
d (2?) d
|
_(B j )(x, !) _(B & j )(x, !) d! dx,
q(x, !)<1
and by (0.2) and (0.7),
: j Res(Q &dB j B & j ) j=1
=
d 8(2?) d+2
|
q(x, !)<1
2?
2?
0
0
| |
(b(#(t ; x, !))&b(#(s ; x, !))) 2 ds dt d! dx. sin 2((t&s)2) (1.5)
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SPECTRAL ASYMPTOTICS
509
Set (x$, %)=#(x, !, s), r=t&s. Then ds dt d! dx=ds dr d% dx$, and (1.5) is equal to d 8(2?) d+2 =
2?
|
q(x$, %)<1
1 4(2?) d+1
| | 0
| | X
2? 0
|%| =1
(b(#(r; x$, %))&b(x$)) 2 dr ds d% dx$ sin 2(r2)
|
? 0
(b(#(r ; x$, %))&b(x$)) 2 dr d% - g(x$) dx$. sin 2(r2)
From this, we get the second term in the right in (0.6).
2. CONCLUDING REMARKS We conclude with a few remarks about multidimensional strong Szego limit theorems which are not specifically associated with a Zoll operator on a compact manifold (but are, for the most part, associated with other Zoll type phenomena): 1. Let A be a zeroth order pseudodifferential operator on R d whose Weyl symbol is polyhomogeneous with respect to the homotheties, (x, !) (* 12x, * 12!), and let P n be projection onto the subspace of L 2(R d ) spanned by the Hermite functions of degree n. Then trace (P n AP n ) s admits a two-tiered asymptotic expansion of the form (1.1) in which Q is the harmonic oscillator. (See [JZ].) 2. Let B d be the closed unit ball in C d and H 2 the space of L 2 holomorphic functions on B d. Given f # C (B d ) let T f be the contraction to H 2 of the operator multiplication by f, and let P n be the projection of H 2 onto the space spanned by the polynomials, : a : z :,
z=(z 1 , ..., z d ).
|:| n
Then trace (P n T f P n ) s admits a two-tiered asymptotic expansion of the form (1.1) involving ``Toeplitz'' analogues of the residues (0.4) (cf. [Gu, 2]) in particular the replacement for Q in (0.4) is the operator, i z r z r .) 3. For d=1 the example above yields the classical Szego theorem (0.1). Both of the previous examples are special cases of a Szego theorem for multidimensional Toeplitz operators of the type discussed in [BG]. (The first example can be translated into a theorem of this type by means of the standard isomorphism between L 2(R d ) and the Bargmann space of holomorphic functions on C d which are L 2 summable with respect to the measure, exp( & |z| 2 ) dz dz.) We will discuss these and other Toeplitz analogues of the theorems in this paper in a future article.
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GUILLEMIN AND OKIKIOLU
4. There is an analogue of the Szego theorem (0.1), where the circle S 1 is replaced by the interval [ &1, 1], and P n is the orthogonal projection onto the polynomials of degree at most n. Here, changing the measure on [ &1, 1] gives rise to different projections P n . Special cases are the Szego theorems for ultraspherical polynomials. (See [Hi].) In [Ok, 2], still working on the interval, P n is the spectral projection for a general Sturm Liouville operator with Dirichlet or Neumann boundary conditions. In [J1] an analogue of (0.1) is proved where S 1 is replaced by a general smooth Jordan curve in the complex plane, and P n is the orthogonal projection onto the polynomials in z of degree at most n, with respect to arc length. 5. If a compact Lie group, K, acts on X and this action preserves Q, there are various equivariant versions of Theorem 0.1. For instance let X=S 2 and let S 1 act on S 2 by rotation about the z-axis. Let A be an S 1 invariant zeroth order pseudodifferential operator and let 1
1 : L 2(S 2 ) L 2(S 2 ) S 1
be orthogonal projection. (Here, L 2(S 2 ) S denotes the S 1 invariant functions in L 2(S 2 ).) Then there is a strong Szego theorem for trace (1P nAP N 1 ) s (which is basically just the Szego theorem for Legendre polynomials first proved in [Hi]). More generally for arbitrary K and X, suppose the action of K on X has the property ( V ) below: For x # X let N x be the co-normal space to K } x at X and let K x be the stabilizer group of x in K. (Note that K x acts linearly on N x .) For all x # X and all ! # N x &0, the stabilizer group of ! in K x is finite. (V) This property implies that the projection operator 1: L 2(X) L 2(X) K is a Fourier integral operator, and one can deduce from this fact a two-tiered Szego asymptotics for trace (1P n AP n 1 ) s analogous to (1.1). 6. Several years ago one of us proved that for any positive self-adjoint first order elliptic pseudodifferential operator, Q: C (X) C (X), one has a weak Szego limit theorem trace(P * AP * ) k =(2?) &d * d
|
a(x, !) k dx d!+o(* d ),
(2.1)
q(x, !)<1
P * being projection onto the eigenfunctions of Q with eigenvalue <*. (See [Gu, 1].) Moreover Hormander ([Ho, 1], Section 29.1) showed that the
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SPECTRAL ASYMPTOTICS
511
o(* d ) on the right can be replaced by O(* d&12 ), and recently Laptev and Safarov [LS] have shown that, in fact, one can replace o(* d ) by O(* d&1 ). Moreover, it is not hard to show that for k=1 in (2.1), the error term on the right is o(* d&1 ) providing the set of periodic trajectories in T*X&O of the bicharacteristic flow associated with q(x, !) is of measure zero (in other words providing Q is non-Zoll in a rather strong sense). These results suggest that there may be two-tiered Szego limit theorems for other operators besides Zoll operators; and, in fact, there are, as the following two examples show. 7. There is an analogue of Lemma 1.1 where X is replaced by R d, A is a pseudo-differential operator on R d of sufficiently negative order, and for any fixed domain 0 in R d, P * is multiplication on the Fourier transform side by the characteristic function of *0. (See [W]). 8. Similar to the case described in 7, there is an analogue of Theorem 0.3 where X is replaced by the flat d-dimensional torus, and for any fixed smooth domain 0 in R d satisfying certain weak conditions, P * is truncation operator on the Fourier transform side to the finite Fourier series supported on *0. (See [Do], [L], [Ok, 1].) Kurt Johansson [Jo, 2] noticed recently that there is a remarkable connection between the asymptotics (0.1) and the random behavior of n_n unitary matrices as n goes to infinity. 1 It would be interesting to know whether the multidimensional Szego theorems described here have similar implications in random matrix theory.
APPENDIX We will give here details of the proof of Lemma 2. We begin by observing that if the bicharacteristic flow on T*X&0 associated with q(x, !) is simply periodic of period 2? then &
trace(exp itQ) At :
c k(A) / k(t)
(1)
k=d&1
where / k(t)= : n ke int.
(2)
n>0 1 He used this observation to settle a conjecture of DiaconisShahshahani which says that the random variables, ( j) &12 trace M j, tend, in a very strong sense, to Gaussian i.i.d.s as n goes to infinity.
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(Indeed, by [DG], Section 4, the left hand side of (1) is a classical co-normal distribution on the circle with wave front set at t=0, {>0, and every such distribution admits an asymptotic expansion of the form above.) Rewriting the left hand side in the form : e itn trace ? n A? n and comparing Fourier coefficients one deduces &
trace ? n A? n t :
c k(A) n k
(3)
k=d&1
We will now show that the c k(A)'s can also be characterized as symbolic residues: Consider the zeta function ` A(z)=trace Q zA. This function has simple poles at z=&d, &d+1, &d+2, etc., and we will show that the residues at these poles are c d&1(A), c d&2(A), etc. (In other words, c k(A)=Res(Q &k&1A).) We will deduce this result from the following more general result: Let Q be any positive self-adjoint first order elliptic 9DO of order one. (i.e. not necessarily an operator of Zoll type.) By [Ho, 1], Section 4 there is an asymptotic expansion of the form &
trace(exp itQ) At :
c k(A) / k(t)
(4)
k=d&1
near t=0 where / k(t)=
|
t ke its ds,
* 0 >0.
(5)
*0
(This / k is not the same as the / k defined by (2), however; it differs from it by a function which is smooth in a neighborhood of zero; so the asymptotic expansion (4) is identical with the asymptotic expansion (1). Theorem. The c k(A)'s occurring in (4) are the residues at the points, z=&k&1, of the zeta function ` A(z)=trace Q zA.
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SPECTRAL ASYMPTOTICS
513
Proof. Let P * be projection onto the span of the eigenfunction of P of eigenvalue * i *, and let _(*)=trace P * AP * . Then @ d_ def (t)=trace(exp itQ) A = e(t). d* Let \ be a function whose Fourier transform is C , is supported in the set |t| <= and is one on the set |t| <=2. For small enough = \^
& @ d_ (t)t : c k(A) /^ k(t); d* k=d&1
so by taking inverse Fourier transforms: & d_ t : c k(A) * k d* k=d&1
\V
(6)
for *> >0. Consider now the zeta function above, ` A(z)=
|
*z
*0
d_ d*, d*
(7)
where 0<* 0
|
*z
*0
\
d_ d_ &\ V d* d* d*
+
(8)
is holomorphic in z in the whole z plane. In other words
|
*z
*0
d_ d* d*
differs from
|
\
*z \ V
*0
d_ d* d*
+
(9)
by a holomorphic function of z. Thus if one substitutes the expansion (6) into (9) one gets &N
` A(z)= : k=d&1
c k(A)
|
* z+k d*+ } } }
1
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(10)
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GUILLEMIN AND OKIKIOLU
where
|
* z+k d*=
1
d z+k+1 1 * z+k+1 d*
}
1
1 =& z+k+1
and the `` } } } '' is a function which is holomorphic for Re z
d _(*) d*
it suffices to show that
|
/ z (*)( f(*)&\ V f (*)) d*
&
is holomorphic in the whole z plane. Therefore, if one rewrites this expression as
|
/$ s(t)(1&\^(t)) f (t) dt
&
and notes that f (t) is a tempered distribution in t, it suffices to prove. Lemma. Proof.
K
(1&\^ ) /$ z is an entire function of z (with values in S(R).) By definition of the inverse Fourier transform /$ z (t)=(2?) &1
|
e i*t#(*) * z d*
&
=(2?) &1
i t
k
\+ |
e i*t
&
d d*
k
\ + #(*) * d* z
Since 1&\^ is zero when |t| <=2, one can show from this that (1&\^ ) /$ z is bounded in all the semi-norms d dt
s
} \ + f}
| f | r, s =sup t r
uniformly on compact subsets of the z-plane. K
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SPECTRAL ASYMPTOTICS
515
Note added in proof. After submitting this paper for publication, we noticed that this second-order asymptotic formula could be replaced by an asymptotics to all orders. For a brief description of this result, which depends very much on the results below, see the research announcement [GO].
REFERENCES [BG] [CV] [Do] [DG] [E] [GS] [Gu, 1] [Gu, 2] [GO] [Hi] [Ho, 1] [Ho, 2] [Jo, 1]
[Jo, 2] [JZ] [K] [LS] [Li] [Ok, 1] [Ok, 2]
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GUILLEMIN AND OKIKIOLU
[Ok, 3] K. Okikiolu, An analogue of the strong Szego limit theorem for a SturmLiouville operator on the interval, preprint. [U] A. Uribe, A symbol calculus for a class of pseudodifferential operators on S n and band asymptotics, J. Funct. Anal. 59 (1984), 535556. [W] H. Widom, ``Asymptotic Expansions for Pseudodifferential Operators in Bounded Domains,'' Lect. Notes in Math., Vol. 1152, Springer-Verlag, BerlinNew York, 1985.
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